Cyclotomic character

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ : G → Aut_R(R(1)) ≈ GL(1, R)).

p-adic cyclotomic character

If p is a prime, and G is the absolute Galois group of the rational numbers, the p-adic cyclotomic character is a group homomorphism

\chi _{p}:G\rightarrow \mathbf {Z} _{p}^{\times }

where Z_p^× is the group of units of the ring of p-adic integers. This homomorphism is defined as follows. Let ζ_n be a primitive pⁿ root of unity. Every pⁿ root of unity is a power of ζ_n uniquely defined as an element of the ring of integers modulo pⁿ. Primitive roots of unity correspond to the invertible elements, i.e. to (Z/pⁿ)^×. An element g of the Galois group G sends ζ_n to another primitive pⁿ root of unity

\theta =\zeta _{n}^{a_{g,n}}

where a_g,n ∈ (Z/pⁿ)^×. For a given g, as n varies, the a_g,n form a comptatible system in the sense that they give an element of the inverse limit of the (Z/pⁿ)^×, which is Z_p^×. Therefore, the p-adic cyclotomic character sends g to the system (a_g,n)_n, thus encoding the action of g on all p-power roots of unity.

In fact, $\chi _{p}$ is a continuous homomorphism (where the topology on G is the Krull topology, and that on Z_p^× is the p-adic topology).

As a compatible system of ℓ-adic representations

By varying ℓ over all prime numbers, a compatible system of ℓ-adic representations is obtained from the ℓ-adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol ℓ to denote a prime instead of p). That is to say, χ = { χ_ℓ }_ℓ is a "family" of ℓ-adic representations

\chi _{\ell }:G_{\mathbf {Q} }\rightarrow \operatorname {GL} _{1}(\mathbf {Z} _{\ell })

satisfying certain compatibilities between different primes. In fact, the χ_ℓ form a strictly compatible system of ℓ-adic representations.

Geometric realizations

The p-adic cyclotomic character is the p-adic Tate module of the multiplicative group scheme G_m,Q over Q. As such, its representation space can be viewed as the inverse limit of the groups of pⁿth roots of unity in Q.

In terms of cohomology, the p-adic cyclotomic character is the dual of the first p-adic étale cohomology group of G_m. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of H²_ét( P¹ ).

In terms of motives, the p-adic cyclotomic character is the p-adic realization of the Tate motive Z(1). As a Grothendieck motive, the Tate motive is the dual of H²( P¹ ).[1]

Properties

The p-adic cyclotomic character satisfies several nice properties.

It is unramified at all primes ℓ ≠ p (i.e. the inertia subgroup at ℓ acts trivially).
If Frob_ℓ is a Frobenius element for ℓ ≠ p, then χ_p(Frob_ℓ) = ℓ
It is crystalline at p.

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References

Section 3 of Deligne, Pierre (1979), "Valeurs de fonctions L et périodes d'intégrales" (PDF), in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics (in French), 33.2, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022

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[1] Section 3 of Deligne, Pierre (1979), "Valeurs de fonctions L et périodes d'intégrales" (PDF), in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics (in French), 33.2, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022